This came out of a discussion with Wyrd Smythe in a comment section on Students showing up at college understanding the fact value distinction is a good thing.
It started as a discussion about morals, but this aside developed in the conversation.
Wyrd said this,
What about the fact that we nowhere find a perfect circle, but we do find imperfect circles that lead us to the idea of a perfect circle? In general, nowhere do we find actual mathematics in nature, but we find everywhere facts that lead to mathematics?
And my response is here, edited for better context here:
We never see perfect abstract things: points, circles, cubes, …, ‘chair’. Even when we do the math we only work with approximations – significant figures, or symbols that represent some fictitious notion. Models with many terms even drop less significant terms as intentional approximations, in order to simplify the models, and then, wow, some complex system nearly fits reality.
The Platonic Forms ideas is all wrong. Material objects are not crude examples of pure platonic forms. Instead, idealised models are approximations to real forms. They are simplistic models, compared to an accurate model of any particular real object. And since all real objects are unique (I think) the ideal models are like statistical models – analogous to the ideal gas laws, for example.
I think the platonic perception comes from the human brain history: we awake and become self aware, and aware of our thinking, but at that time we know nothing of evolution. We come ready made. The material world seems messy, and the mental world seems cleaner, more perfect. And we can imagine things we don’t see. So the mental world is an indication of a more perfect reality. The primacy of the mental over the physical is established, so the imagined pure things must be real. But that turns out to be fiction, dreams, imagination, idealistic approximations. We are no more than evolved complex chemistry, sloppy biology, that survives.
Why should we expect the universe to be pure or perfect in any way? Why should we think there’s any sort of reality to perfect models?
Wyrd responded here. And what follows addresses his view as presented, but expands on mine.
“In fact, any real world circle is the approximation of the abstraction of a circle.”
I disagree because abstract circles don’t exist except as conceptual patterns in minds, as sometimes expressed by mathematical correlates. And, since minds are really a brain’s eye view of itself, and brains at various levels are quantised, as synapses, action potentials, molecules, atoms, then there is nowhere where an abstract circle exists. The mathematics of circles happen to fit well with the imaginary abstract circles, and not quite so well with real objects that we call circles.
At the very best, one could say that abstract circles and real circles are relative approximates to each other.
…it’s very difficult to imagine an intelligent race failing to discover a theory of circles. A circle is easy to define.
Yes, I can imagine aliens would figure out a theory of circles, and symbolism aside I would guess it would be the same as ours, or in some for such that the two theories could be derived from each other, perhaps starting with different axioms. But, how would aliens come to a theory of circles if they didn’t actually have any circles to play with. Which came first, pi.r^2, or wheels? Or was it logs? How do brains come to any knowledge without experience. Even what appear to innate capacities, to distinguish between horizontal and vertical lines to different degrees, to recognise certain parts of the electromagnetic spectrum, and acquired experiences passed on through inheritance. In what sense do abstractions exist if a brain hasn’t had some experience that triggers some pattern.
One might wonder about abstractions that are never experienced in actuality – flights of fancy, original art work. The representation, as patterns in the brain, is the real world ‘thing’ that represents the abstraction. That it is a pattern corresponding to nothing in the real world is irrelevant. The abstraction doesn’t exist still; it’s the brain pattern that exists.
Collecting a mass of data and calculating the mean is the act of producing an abstract measure that is use as an approximate representation for the bulk of the data. That same mean value could be the mean of countless slightly different distributions of data from real world things.
Any abstract circle is an approximate representation of many real circles that do or could exist or could be created. It’s a representation of real things. That a mathematical theory works well is an interesting fact. But that in itself does not prove of demonstrate that real things are approximations to abstractions rather than the other way round.
My reason for preferring to use the other perspective is that moves away from the mistaken reification of abstract ideas as pure or transcendent, which are all too often taken to be reality.
“The natural abstraction necessary to enable counting…”
I rather think it’s the case that dealing with objects came first, as an empirical experience, before developing counting. Other animals seem to get on quite well without such abstractions, and surely without abstract maths.
Animal brains that can detect moving objects, that can tell when they are outnumbered, outsized, are processing sense data that does not need an abstraction theory that it can contemplate outside the moment is is using it’s skills. I’m guessing, but I don’t think a lion sits quietly contemplating how better to get the next prey animal, musing on predator-prey motion theory, because today’s prey managed to escape it. And creatures with no nervous system don’t do any brain stuff at all – they are essentially constrained phsyics and chemistry automata. The ‘mathematical’ nature of the universe is something we put on it. It conforms to patterns because of the nature of reality. Aliens will discover circles when they look up at the sky and see a sun or a moon. At a distance they will look like perfect circles, because their eyes cannot see the real edges formed by the horizons.
Having said that, I’m not dismissive of all abstract ideas. But the ones I see having potential as an aspect of reality are quite different. Of course there’s Wheeler’s ‘it from bit’ and Luciano Floridi’s ‘information’, which abstracts away the whole of reality – but they are reductionist views on our reality not separate abstractions to which reality conforms. There are no circles in those abstract perspectives as far as I’m aware – there’s not much of anything.
But proper calculation requires an analog computer with a complexity order equal to the real world. In fact, the real world is that analog computer; it calculates the three-body problem and weather predictions with ease.
Well, it could be digital – the old atomist argument. But even so, circles are constructs in that reality, not a part of it. The reduction of reality to its constituents does not necessarily show us any circles down there, so the maths of circles is still just models of how that lower scale reality builds up into real atoms and real circles made of lumpy atoms.
Note that I’m not claiming Plato’s realm of perfect forms has ontological reality, but I think it has an almost undeniable epistemological reality. If you grant that any intelligent species will discover mathematics and geometry, it almost has to have it.
Ghosts have epistemological reality. We can ‘know’ about them conceptually. But they are not real, as far as I know. This is partly why I’m not convinced by Platonic forms and other neat abstractions like circles. We can actually have imaginative abstractions of messy things too. And things that don’t exist. That seems a clue to me that all abstractions do not exist, and that they are creations in the mind that we invent, and are sometimes approximate representations of reality.
And, you can also say that any species that deals with the real world and real objects will invent abstract approximations to make life easier.
If I design a building, but don’t build it, is the building real? Isn’t there some form of reality to the design?”
Yes. It consists first as messy vague patterns in neurons. It’s committed to paper by the messy accumulation of real pencil lines (or as quantised bits, bytes, words, numeric finite representations in a computer). Nowhere ever does it exist as what we abstract as ‘the design’. We pretend it does. Using the term, “The design”, is no more than using a mathematical symbol to represent a complex expression. Brains that review the messy pencil lines (or printout, or digital screen display, or ‘analogue’ electron display) only ever hold vague mental representations. We juggle parts of the design around, focus on bits, even use language to express numbers in the design. Photocopy the ‘design drawing’ and the ink patterns will differ, copy the data to a different computer and it will be represented still by quantised bits.
The abstraction of the design is no more real than gods.
In fact ‘information’ is an interesting ‘thing’. It doesn’t exist. So, how is information ‘transferred’? I’ll finish with a couple of examples of that.
It’s an abstract approximation to reality. It’s a representation of often quite distinct things that have some correspondence.
Example: Computer Information
When I download an ‘app’ onto my phone, there is nothing on the server computer that arrives on my phone. Nothing. The server computer sends electrons in patterns around its circuits, reading bits from memory. They are used to induce changes in a transmission port that takes electrons from a power source. The pattern of the transmitted electrons down the wire has some correspondence with the patter of electrons representing the app in the server’s memory. The app in the server’s memory doesn’t go anywhere – well, depending on the type of memory: dynamic memory is refreshed, so that the bit pattern representing the app is maintained. In reality that pattern may get shifted around, destroyed and recreated – all depending on the state of the system and its management of data.
In the meantime electrons are going down wires, inducing transformations in yet other wires by transducers. There may be a satellite hop involved eventually, or maybe a microwave link to your local phone cell. None of the original atoms or electrons or energy are present by this time. Power supplies along the way are providing energy, electrons, heat, to make local patterns that correspond to the incoming patterns, and in turn cause outgoing patterns to match.
Eventually electromagnetic signals induce electron flow in my phone’s antenna, circuits decode the data, patterns are created in my phone that match the app pattern on the distant server.
And there is nothing on my phone that was on the server. Nothing. My phone merely contains a pattern of bits that correspond to a pattern of bits somewhere else.
Think about the energy. All the energy used in my phone comes from its battery. The same for millions of phones that have ‘downloaded’ the app. There is not some drain like a tap on a barrel when an app is downloaded. There is a drain, or course, but very little of the energy goes out through the wires in the server network – we’re talking volts and milliapms, and nearly all that comes from a power feed into the output chips, not from ht memory bits where the app is stored as a bit pattern. The energy used by a server mostly leaves as heat.
And that term, ‘download’. We’re used to the term ‘download’, but it’s a metaphor. In reality patterns are copied. In the transmission process the ‘information’ (that is never an ontological ‘thing’) goes through many encodings and decodings. Even the sequence of bits can be out of order because the ‘information’ is transferred in packets, which may be lost and resent.
Example: Shouting across a field.
You and a friend stand at opposite ends of a field. You read aloud a list of common names, and he has to remember them and shout them back, in order. No matter, no energy, that left your mouth, reaches his ears. Sequences of colliding air atoms absorb and retransmit energy, dissipating some as heat, passing some on a motion in the next atom. Much of that energy is transmitted radially, so what reaches your friend’s ear is a tiny representation of the sound that left your mouth. His ear picks it up. None of the air that entered his ear, vibrating the hairs in his ear, none of it ends up in his auditory cortex.
We say you have given him some information. It’s meaningful information to him because his brain has become accustomed to the neuron signals, the patterns of firings, entering his auditory cortex when names are heard. Without that language context all that’s going on in his brain is some neurons are firing, triggering ready made patterns, doing what brains do when they recognise stuff.
No matter or energy that left your brain as you read and spoke these names, none of it entered his head.
Information is an abstract term we use to describe that process. We think of information as a ‘thing’ because as physical creatures that have ancestors that didn’t have brains, when we experienced something it was all physical, chemical, biological. It still is, but eyes, ears, voices, they allow us to make remote physical connections. We don’t need to ‘transfer’ ‘information’ directly, through touch, we do it indirectly, through the touch of air movement in our ears, or the arrival of photons in our eyes.